Headlights both quit working at same time,fuses and lamps are good,brights work fine.2002 buick century. is there a relay or something in common with both lights?

Sounds like two separate problems. For the headlight, check the headlight relay in the underhood fuse box (may be more than one if you have DRL), but may also be the headlight switch. For the dash lights, there may be a dead spot on the brightness control. Rotate it a few time to see if that fixes the problem.

Try checking the fusebox under the engine compartment. You should find fuses for both lights in there. also check your light relays also in the same compartment. You might have a bad relay.

Your high beam switch is bad most likely.

there is a relay under the hood in the fuse/relay panel called drl, or daytime running lamps, that relay is most likely bad, a good way to tell is switch it with the high beam relay, if they come on you know thats it, if they dont, its the relay for the lowbeams, again switch it with the high beam relay and see.. either way i can tell you for sure one of those two are bad... but most of the time it is the daytime running lamps, or sometimes called driving lights relay. hope this helps you

There could be a timed circuit breaker:

Circuit Breakers The headlights are protected by a circuit breaker in the headlamp switch. If the circuit breaker trips, the headlights will either flash on and off, or stay off altogether. The circuit breaker rests automatically after the overload is removed. The windshield wipers are also protected by a circuit breaker. If the motor overheats, the circuit breaker will trip, remaining off until the motor cools or the overload is removed. One common cause of overheating is operation of the wipers in heavy snow. The circuit breakers for the power door locks and power windows are located in the fuse box.

where is the location of the headlamp relay on a 2001 Buick century

there is a inst lamp fuse, is it OK? the headlight switch controls this. most likely it wili need the isnstrumen dimmer switch replaced.... if the fuses are good and are getting power when the headlight switch is on.

Go through all your fuses, Dim lights run on a separate fuse.

Check the headlight fuses in the engine fuse box compartment. Not sure if the lights have relays, but they will be clearly mark in the inside of the top cover for location.